An advantage of \(q\) -variables is that higher-order accuracy can be achieved even at boundary points as the defect-correction procedure can be combined with the kinetic wall [1]} and kinetic outer boundary [2]} conditions. Furthermore, the distribution functions \(F\left(\widetilde{q}_i \right) \)  and \(F\left(\widetilde{q}_0 \right) \)  corresponding to \(\widetilde{q}_i\)  and \(\widetilde{q}_0\)  are always Maxwellians and therefore preserves the positivity of numerical solution.
